On generalized inverses of singular matrix pencils

 Access to full text

 Full (PDF)

1. Ayasun, S., Nwankpa, C.O. and Kwatny, H.G. (2004). Computation of singular and singularity induced bifurcation points of differential-algebraic power system model, IEEE Transactions on Circuits and Systems I 51(8): 1558. Zbl1082.65558
2. Bandler, J.W. and Zhang, Q.J. (1986). Large change sensitivity analysis in linear systems using generalized householder formulae, International Journal of Circuit Theory and Applications 14(2): 89-101. 
3. Beelen, T. and van Dooren, P. (1988). An improved algorithm for the computation of Kronecker's canonical form of a singular pencil, Linear Algebra and Its Applications 105: 9-65. Zbl0645.65022
4. Ben-Israel, A. and Greville, T.N.E. (1974). Generalized Inverses: Theory and Applications, Wiley-Interscience, New York, NY. Zbl0305.15001
5. Boullion, T.L. and Odell, P.L. (1971). Generalized Inverse Matrices, Wiley-Interscience, New York, NY. Zbl0223.15002
6. Brenan, K.E., Campbell, S.L. and Petzold, L.R. (1996). Numerical Solution of Initial-Value Problems in DifferentialAlgebraic Equations, 2nd Edn., SIAM, Philadelphia, PA. Zbl0844.65058
7. Bunse-Gerstner, A., Byers, R., Mehrmann, V. and Nichols, N.K. (1991). Numerical computation of an analytic singular value decomposition of a matrix valued function, Numerische Mathematik 60(1): 1-39. Zbl0743.65035
8. Campbell, S.L. and Meyer, C.D. (1979). Generalized Inverses of Linear Transformations, Dover Publications, New York, NY. Zbl0417.15002
9. Davies, A.C. (1966). Topological solution of networks containing nullators and norators, Electronics Letters 2(3): 90-92. 
10. Demmel, J. and Kågström, B. (1993). The generalized Schur decomposition of an arbitrary pencil λA-B, ACM Transactions on Mathematical Software 19(2): 160-174. 
11. Dziurla, B. and Newcomb, R. (1989). Input-output pairing in LTV semistate systems, IEEE Transactions on Circuits and Systems 36(1): 139-141. 
12. Dziurla, B. and Newcomb, R.W. (1987). Nonregular semistate systems: Examples and input-output pairing, Proceedings of the IEEE Conference on Decision and Control, Los Angeles, CA, USA, pp. 1125-1126. 
13. Estévez Schwarz, D. and Tischendorf, C. (2000). Structural analysis of electric circuits and consequences for MNA, International Journal of Circuit Theory and Applications 28(2): 131-162. Zbl1054.94529
14. Fosséprez, M. (1992). Non-linear Circuits-Qualitative Analysis of Non-linear, Non-reciprocal Circuits, Wiley, Chichester. 
15. Gantmacher, F.R. (1959). Theory of Matrices, Vol. II, Chelsea, New York, NY. Zbl0085.01001
16. Gear, C.W. (1971). Simultaneous numerical solution of differential-algebraic equations, IEEE Transactions on Circuit Theory 18(1): 89-95. 
17. Gear, C.W. and Petzold, L.R. (1982). Differential/algebraic systems and matrix pencils, in B. Kågström and A. Ruhe (Eds.), Matrix Pencils, Lecture Notes in Mathematics, Vol. 973, Springer-Verlag, New York, NY, pp. 75-89. Zbl0494.65038
18. Gear, C.W. and Petzold, L.R. (1984). ODE methods for the solution of differential/algebraic systems, SIAM Journal on Numerical Analysis 21(4): 717-728. Zbl0557.65053
19. Griepentrog, E. and März, R. (1986). Differential-Algebraic Equations and Their Numerical Treatment, Teubner-Texte zur Mathematik, Vol. 88, Teubner Verlagsgesellschaft, Leipzig. Zbl0629.65080
20. Günther, M. and Feldmann, U. (1999a). CAD-based electriccircuit modeling in industry, I: Mathematical structure and index of network equations, Surveys on Mathematics for Industry 8: 97-129. Zbl0923.65039
21. Günther, M. and Feldmann, U. (1999b). CAD-based electriccircuit modeling in industry, II: Impact of circuit configurations and parameters, Surveys on Mathematics for Industry 8: 131-157. Zbl0923.65040
22. Hairer, E., Lubich, C. and Roche, M. (1989). The Numerical Solution of Differential-Algebraic Systems by RungeKutta Methods, Lecture Notes in Mathematics, Vol. 1409, Springer-Verlag, Berlin. Zbl0683.65050
23. Hasler, M. (1986). Non-linear non-reciprocal resistive circuits with a structurally unique solution, International Journal of Circuit Theory and Applications 14(3): 237-262. Zbl0621.94022
24. Hayton, G.E., Pugh, A.C. and Fretwell, P. (1988). Infinite elementary divisors of a matrix polynomial and implications, International Journal of Control 47(1): 53-64. Zbl0661.93016
25. Hou, M. (1995). Descriptor Systems: Observers and Fault Diagnosis, Fortschrittsberichte, Reihe 8: Meß-, Steuerungsund Regelungstechnik, Vol. 482, VDI Verlag, Düsseldorf. 
26. Hou, M. and Müller, P.C. (1992). A singular matrix pencil theory for linear descriptor systems, Proceedings of the Symposium on Implicit and Nonlinear Systems, Ft Worth, TX, USA, pp. 178-190. 
27. Hou, M., Pugh, A.C. and Hayton, G.E. (1997). Generalized transfer functions and input-output equivalence, International Journal of Control 68(5): 1163-1178. Zbl0891.93010
28. Karcanias, N. (1987). On the characteristic, Weyr sequences, the Kronecker invariants and canonical form of a singular pencil, in R. Isermann (Ed.), Preprint from Automatic Control World Congress, Pergamon Press, Munich, pp. 109-114. 
29. Karcanias, N. and Hayton, G.E. (1981). Generalized autonomous dynamical systems, algebraic duality and geometric theory, in H. Akashi (Ed.), Preprint from Automatic Control World Congress, Pergamon Press, Kyoto, pp. 289-294. 
30. Kronecker, L. (1890). Algebraic reduction of pencils of bilinear forms, Sitzungsberichte der Preussischen Akademie der Wissenschaften, pp. 1225-1237, (in German). Zbl22.0169.01
31. Kublanovskaya, V.N. (1983). Analysis of singular matrix pencils, Journal of Mathematical Sciences 23(1): 1939-1950. Zbl0515.65031
32. Kunkel, P. and Mehrmann, V. (1990). Numerical solution of differential algebraic Riccati equations, Linear Algebra and Its Applications 137/138: 39-66. Zbl0707.65043
33. Kunkel, P. and Mehrmann, V. (2006). Differential-Algebraic Equations: Analysis and Numerical Solution, EMS Publishing House, Zürich. Zbl1095.34004
34. Kunkel, P., Mehrmann, V., Rath, W. and Weickert, J. (1997). A new software package for linear differentialalgebraic equations, SIAM Journal on Scientific Computing 18(1): 115-138. Zbl0868.65041
35. Kwatny, H.G., Fischl, R.F. and Nwankpa, C.O. (1995). Local bifurcation in power systems: Theory, computation, and application, Proceedings of the IEEE 83(11): 1456-1483. 
36. Marszalek, W. and Trzaska, Z.W. (2005). Singularity-induced bifurcations in electrical power systems, IEEE Transactions on Power Systems 20(1): 312-320. 
37. Mehrmann, V.L. (1991). The Autonomous Linear Quadratic Control Problem, Lecture Notes in Control and Information Science, Vol. 163, Springer-Verlag, Berlin. Zbl0746.93001
38. Meyer, C.D. and Rose, N.J. (1977). The index and the Drazin inverse of block triangular matrices, SIAM Journal on Applied Mathematics 33(1): 1-7. Zbl0355.15009
39. Müller, P.C. (2005). Remark on the solution of linear timeinvariant descriptor systems, Proceedings in Applied Mathematics and Mechanics 5(1): 175-176. 
40. Özcaldiran, K. and Lewis, F.L. (1990). On the regularizability of singular systems, IEEE Transactions on Automatic Control 35(10): 1156-1160. Zbl0724.93011
41. Pandolfi, L. (1981). On the regulator problem for linear degenerate control systems, Journal of Optimization Theory and Applications 33(2): 243-254. Zbl0421.93036
42. Penrose, R. (1955). A generalized inverse for matrices, Proceedings of the Cambridge Philosophical Society 51: 406-413. Zbl0065.24603
43. Rao, C.R. and Mitra, S.K. (1971). Generalized Inverse of Matrices and Its Applications, John Wiley & Sons, New York, NY. Zbl0236.15004
44. Reinschke, K. and Schwarz, P. (1976). Methods for ComputerAided Analysis of Linear Networks, Elektronisches Rechnen und Regeln, Vol. 9, Akademie-Verlag, Berlin, (in German). 
45. Reißig, G. (1996). Differential-algebraic equations and impasse points, IEEE Transactions on Circuits and Systems I 43(3): 122-133. 
46. Reißig, G. (1998). Contributions to the Theory and Applications of Implicit Differential Equations, Ph.D. thesis, Technische Universität Dresden, Dresden, (in German). Zbl0936.34001
47. Reißig, G. (1999). Extension of the normal tree method, International Journal of Circuit Theory and Applications 27(2): 241-265. Zbl0921.68004
48. Reißig, G. and Boche, H. (2003). On singularities of autonomous implicit ordinary differential equations, IEEE Transactions on Circuits and Systems I 50(7): 922-931. 
49. Reißig, G. and Feldmann, U. (1996). Computing the generic index of the circuit equations of linear active networks, Proceedings of the International Symposium on Circuits and Systems, Atlanta, GA, USA, Vol. III, pp. 190-193. 
50. Riaza, R. (2004). A matrix pencil approach to the local stability analysis of non-linear circuits, International Journal of Circuit Theory and Applications 32(1): 23-46. Zbl1037.94556
51. Riaza, R. (2006). Time-domain properties of reactive dual circuits, International Journal of Circuit Theory and Applications 34(3): 317-340. Zbl1123.94410
52. Riaza, R. (2008). Differential-Algebraic Systems: Analytical Aspects and Circuit Applications, World Scientific, River Edge, NJ. Zbl1184.34004
53. Röbenack, K. (1999). Contribution to the Analysis of Descriptor Systems, Shaker-Verlag, Aachen, (in German). Zbl0947.93506
54. Röbenack, K. and Reinschke, K.J. (1998). Digraph based determination of Jordan block size structure of singular matrix pencils, Linear Algebra and Its Applications 275-276: 495-507. Zbl0934.15012
55. Röbenack, K. and Reinschke, K.J. (2000). Structural analysis of the input-output behaviour of singular descriptor systems, Workshop über Deskriptorsysteme, Paderborn, Germany, (in German). Zbl0966.93027
56. Sannuti, P. (1981). Singular perturbations in the state space approach of linear electrical networks, International Journal of Circuit Theory and Applications 9(1): 47-57. Zbl0455.94028
57. Sincovec, R.F., Erisman, A.M., Yip, E.L. and Epton, M.A. (1981). Analysis of descriptor systems using numerical algorithms, IEEE Transactions on Automatic Control 26(1): 139-147. Zbl0495.93027
58. Straube, B., Reinschke, K., Vermeiren, W., Röbenack, K., Müller, B. and Clauß, C. (2001). DAE-index increase in analogue fault simulation, in R. Merker and W. Schwarz (Eds.), System Design Automation-Fundamentals, Principles, Methods, Examples, Kluwer, Boston, MA, pp. 221-232. 
59. Tischendorf, C. (1996). Graph-theoretic determination of the structural index of algebraic-differential equations in network analysis in W. Mathis and P. Noll (Eds.), Neue Anwendungen theoretischer Konzepte in der Elektrotechnik mit Gedenksitzung zum 50. Todestag von Wilhelm Cauer, VDE-Verlag, Berlin, pp. 55-60, (in German). 
60. Tischendorf, C. (1999). Topological index calculation of DAEs in circuit simulation, Surveys on Mathematics for Industry 8(3-4): 187-199. Zbl1085.94513
61. van Dooren, P.M. (1981). The generalized eigenstructure problem in linear system theory, IEEE Transactions on Automatic Control 26(1): 111-129. Zbl0462.93013
62. Vardulakis, A.I.G. and Karcanias, N. (1983). Relations between strict equivalence invariants and structure at infinity of matrix pencils, IEEE Transactions on Automatic Control 28(4): 514-516. Zbl0519.93025
63. Vardulakis, A.I.G., Limebeer, D.N.J. and Karcanias, N. (1982). Structure and Smith-MacMillan form of a rational matrix at infinity, International Journal of Control 35(4): 701-725. Zbl0495.93010
64. Varga, A. (1998). Computation of inner-outer factorization of rational matrices, IEEE Transactions on Automatic Control 43(5): 684-688. Zbl0907.65020
65. Varga, A. (2001). Computing generalized inverse systems using matrix pencil methods, International Journal of Applied Mathematics and Computer Science 11(5): 1055-1068. Zbl1031.93071
66. Weierstrass, K. (1868). On the theory of bilinear and quadratic forms, Monatsbericht der Preussischen Akademie der Wissenschaften, reprinted in Mathematische Werke von Karl Weierstrass, Vol. II, 1985, Mayer & Müller, Berlin, pp. 310-338, (in German). Zbl01.0054.04
67. Wilkinson, J.H. (1979). Kronecker's canonical form and the QZ-algorithm, Linear Algebra and Its Applications 28: 285-303. Zbl0458.65022

1. Tomasz Górecki, Maciej Łuczak, Linear discriminant analysis with a generalization of the Moore-Penrose pseudoinverse
2. Predrag Stanimirović, Marko Miladinović, Igor Stojanović, Sladjana Miljković, Application of the partitioning method to specific Toeplitz matrices